Does $f_n$ necessarily converge to f pointwise?

Question

Let $(f_n)_{n=1}^∞$ with $f_n : [a,b] → ℝ$ be a sequence of Riemann integrable functions that converges pointwise to $f : [a,b] → ℝ$. Do we necessarily have

$\int_a^bf_n(x)dx → \int_a^bf(x)dx$?

Answer 1

No ofc not, e.g take $f_n: [0,1] \to \Bbb{R}$ defined by $$f_n := n\cdot1_{\left(0,\frac{1}{n}\right)}$$ which converges pointwise to $f \equiv 0$ but $$\int_0^1 f_n(x) dx = 1$$ for all $n\in \Bbb{N}$

For this to hold you need additional assumptions like dominated convergence or monotone convergence.

Answer 2

No, however, if we add a constraint that there exists a nonnegative Riemann-integrable function $g:[a,b] \to \Bbb{R}$ such that $∣f_n(x)∣≤g(x)$ almost everywhere on $[a,b]$ for all $n∈\Bbb{N}$, then we get the famous Lebesgue's Dominated Convergence Theorem.

(Note that Lebesgue-integrable is a generalization of Riemann-integrable, and as such we can use the theorem for this specific case.)

Also lookup the Monotone Convergence Theorem(s), as those are also quite similar.